Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}6x+5y &= 1 \\ 3x+5y &= -5\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}-6x-5y &= -1\\ 3x+5y &= -5\end{align*}$ Add the top and bottom equations. $-3x = -6$ Divide both sides by $-3$ and reduce as necessary. $x = 2$ Substitute $2$ for $x$ in the top equation. $6( 2)+5y = 1$ $12+5y = 1$ $5y = -11$ $y = -\dfrac{11}{5}$ The solution is $\enspace x = 2, \enspace y = -\dfrac{11}{5}$.